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Classifying modules over \(K\)-theory spectra. (English) Zbl 0911.55005
This paper works in the category of \(S\)-algebras, the brave new world developed recently by May and others, which enables the algebraic topology of spectra to be done at the point-set level. One of the main general results states that if \(R\) is an \(S\)-algebra, then an \(R_*\)-module of projective dimension at most 2 can be realized as the module of homotopy groups of some \(R\)-module, uniquely if this projective dimension is less than 2. This applies to the \(crt\)-category (with objects \(ko\), \(ku\), and \(kt\)), for it is shown that all \(crt\)-acyclic \(crt\)-modules have projective dimension at most 2. For a fixed \(crt\)-module \(M\) of projective dimension exactly 2, there is an equivalence relation finer than homotopy equivalence so that equivalence classes of \(ko\)-modules \(X\) with \(\pi_*^{crt}(X)=M\) correspond to elements of Ext\(_{crt}^{2,-1}(M,M)\). One consequence is that any \(ko_*\)-module \(M_*\) can be realized as the homotopy groups of some \(ko\)-module.

MSC:
55P42 Stable homotopy theory, spectra
55N15 Topological \(K\)-theory
19L41 Connective \(K\)-theory, cobordism
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